From A First Course in Linear Algebra
Version 2.99
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
Summary Linear transformation with a domain smaller that its codomain, so it is guaranteed to not be surjective. Happens to be injective.
A linear transformation: (Definition LT)
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A basis for the null space of the linear transformation: (Definition KLT)
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Injective: Yes. (Definition ILT)
Since T
=
0
A basis for the range of the linear transformation: (Definition RLT)
Evaluate the linear transformation on a standard basis to get a spanning set for
the range (Theorem SSRLT):
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If the linear transformation is injective, then the set above is guaranteed to be linearly independent (Theorem ILTLI). This spanning set may be converted to a “nice” basis, by making the vectors the rows of a matrix (perhaps after using a vector reperesentation), row-reducing, and retaining the nonzero rows (Theorem BRS), and perhaps un-coordinatizing. A basis for the range is:
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Surjective: No. (Definition SLT)
The dimension of the range is 3, and the codomain
(
To be more precise, verify that
2 1 −3 2 6
∉ℛ
T
2 1 −3 2 6
Subspace dimensions associated with the linear transformation. Examine parallels with earlier results for matrices. Verify Theorem RPNDD.
Invertible: No.
The relative dimensions of the domain and codomain prohibit any possibility of
being surjective, so apply Theorem ILTIS.
Matrix representation (Theorem MLTCV):
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